In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.

Will Sawin described the monad structure of this endofunctor in an answer, and in the comments thread Tom Leinster brought up the Giry monad, saying, "it's almost inconceivable that there could be two different reasonable monad structures on this endofunctor, so I bet it's the same."

For the benefit of future researchers, I've created this community wiki thread to aggregate possible applications of the Giry monad in probability and statistics. My hope is that this thread might be a place for the structuralist and probability communities to come together and learn from each other.

If you see any interesting applications of the Giry monad, please post them here.